Average Error: 0.2 → 0.2
Time: 41.0s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3}}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3}}}
double f(double x, double y) {
        double r241537 = 1.0;
        double r241538 = x;
        double r241539 = 9.0;
        double r241540 = r241538 * r241539;
        double r241541 = r241537 / r241540;
        double r241542 = r241537 - r241541;
        double r241543 = y;
        double r241544 = 3.0;
        double r241545 = sqrt(r241538);
        double r241546 = r241544 * r241545;
        double r241547 = r241543 / r241546;
        double r241548 = r241542 - r241547;
        return r241548;
}
↓
double f(double x, double y) {
        double r241549 = 1.0;
        double r241550 = x;
        double r241551 = 9.0;
        double r241552 = r241550 * r241551;
        double r241553 = r241549 / r241552;
        double r241554 = r241549 - r241553;
        double r241555 = 1.0;
        double r241556 = sqrt(r241550);
        double r241557 = y;
        double r241558 = 3.0;
        double r241559 = r241557 / r241558;
        double r241560 = r241556 / r241559;
        double r241561 = r241555 / r241560;
        double r241562 = r241554 - r241561;
        return r241562;
}

Error

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Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{\color{blue}{1 \cdot 3}}}{\sqrt{x}}\]

  6. Applied *-un-lft-identity0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{\color{blue}{1 \cdot y}}{1 \cdot 3}}{\sqrt{x}}\]

  7. Applied times-frac0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\color{blue}{\frac{1}{1} \cdot \frac{y}{3}}}{\sqrt{x}}\]

  8. Applied associate-/l*0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{1}{1}}{\frac{\sqrt{x}}{\frac{y}{3}}}}\]

  9. Final simplification0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3}}}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))